Master game theory with our Nash Equilibrium Calculator. Quickly identify Pure Strategy Nash Equilibria (PSNE) and Mixed Strategy Nash Equilibria (MSNE) for any 2×2 payoff matrix, helping you predict strategic outcomes in economics, business, and social sciences.
Nash Equilibrium Calculator
Enter payoffs for Player 1 (Row) and Player 2 (Column) as (P1, P2):
| Column Left | Column Right | |
|---|---|---|
| Row Top |
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| Row Bottom |
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Nash Equilibrium Calculator Formula
For a 2×2 matrix, a Pure Strategy Nash Equilibrium exists if:
$$\text{Player 1: } U_1(s_1^*, s_2^*) \geq U_1(s_1, s_2^*) \text{ for all } s_1$$
$$\text{Player 2: } U_2(s_1^*, s_2^*) \geq U_2(s_1^*, s_2) \text{ for all } s_2$$
Formula Source: Stanford Encyclopedia of Philosophy
Variables:
- P1 (Player 1 Payoff): The utility or reward received by the row player for a specific outcome.
- P2 (Player 2 Payoff): The utility or reward received by the column player for a specific outcome.
- Strategies: Top/Bottom for Player 1; Left/Right for Player 2.
What is Nash Equilibrium?
The Nash Equilibrium is a fundamental concept in game theory where no player can increase their own expected payoff by changing their strategy while the other players keep theirs unchanged. It represents a state of “strategic stability” in competitive environments.
In a 2×2 game, players may find a single stable outcome (Pure Strategy) or a mathematical probability distribution (Mixed Strategy) where they remain indifferent between their choices, preventing exploitation by the opponent.
How to Calculate Nash Equilibrium (Example)
Let’s use the “Stag Hunt” game as an example:
- Identify Best Responses: If Player 2 plays Left, Player 1 compares Top (3) vs Bottom (0). Top is best.
- Mutual Best Responses: If both players’ choices are best responses to each other in a specific cell, that cell is a Nash Equilibrium.
- Calculate Mixed Strategies: Solve for probabilities $p$ and $q$ that make the opponent indifferent between their strategies.
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Frequently Asked Questions (FAQ)
Can a game have more than one Nash Equilibrium? Yes, games like the Battle of the Sexes or Stag Hunt have multiple Pure Strategy Nash Equilibria.
What is a Mixed Strategy? It is a strategy where a player chooses between possible moves according to a specific probability distribution to remain unpredictable.
Does every game have a Nash Equilibrium? Yes, John Nash proved that every finite game has at least one Nash Equilibrium, either in pure or mixed strategies.
What is a Dominant Strategy? A strategy that is the best choice for a player regardless of what the other player does.